Subtopic: Power transmission systems – gear trains, differential, final drives and power take-off
Question Statement:
A planetary epicyclic reduction gear unit in a tractor wheel hub has a stationary ring (annulus) gear with $Z_r = 72$ teeth and a rotating central sun gear with $Z_s = 18$ teeth driven by the axle shaft. The output drive is taken from the planet carrier. The speed reduction ratio ($i = \frac{N_{\text{sun}}}{N_{\text{carrier}}}$) is ________ (answer in integer).
Answer Key:
5
Step-by-Step Solution & Explanation
```
Epicyclic Planetary Gear Hub Reduction:
Sun Gear (Z_s = 18): Input
Ring Gear (Z_r = 72): Fixed / Stationary
Planet Carrier: Output
```
**Step-by-step Calculation:**
1. Formula for planetary gear set with fixed ring gear:
$\frac{N_s - N_c}{N_r - N_c} = -\frac{Z_r}{Z_s}$
2. Since the ring gear is fixed ($N_r = 0$):
$\frac{N_s - N_c}{0 - N_c} = -\frac{Z_r}{Z_s} \implies \frac{N_s - N_c}{-N_c} = -\frac{Z_r}{Z_s}$
$1 - \frac{N_s}{N_c} = -\frac{Z_r}{Z_s} \implies \frac{N_s}{N_c} = 1 + \frac{Z_r}{Z_s}$
3. Substituting tooth counts:
$\frac{N_s}{N_c} = 1 + \frac{72}{18} = 1 + 4 = 5.0$
Speed reduction ratio is exactly 5.